I was going through this question (Why don't planets have Circular orbits?) related to planetary orbits. In the accepted answer it is stated that orbits are actually conic sections.
Given this understanding, is it possible to find out the shape into which spacetime is warped by a massive object like a Star? Is it anything that we will be able to comprehend from the equations?
If you are asking whether we can calculate the curvature from the planetary orbits then the answer is no. The orbits are curves called geodesics and while you can calculate the geodesics starting from the curvature you cannot reverse the process i.e. start with the orbits and calculate the curvature.
If you are interested in the details this exact issue is discussed in:
(the latter is the more detailed discussion)
The curvature is calculated by solving the Einstein equation. For setups like the Solar system the geometry is very close to a solution of the Einstein equation called the Schwarzschild metric. This gives orbits that are almost but not exactly conic sections. The deviation of the orbits from a conic section is exceedingly small, but it can be measured. For example it is responsible for the anomalous precession of Mercury.
Considering ordinary stars (excluding very high velocity & very high rotation etc.), the curvature is always spherical shape directed towards center of mass. It is downward/inward curvature. We know it from "free fall from initial state of rest always directed towards center of mass".
If initial state is not rest, then the navigation is not directly towards center, it varies depending upon the initial conditions. It can be circular or elliptical, or may crash in the star. Shape of orbit does not impact/determine shape of the curvature. The strength of spherically downward curvature, and velocity and distance of the orbiting body from the star determine shape of the orbit.
We do not need to determine shape of curvature, we know it is spherical.
